Here, it doesn’t really matter what the implicated statement (here, q) is, if the premise (here, p) itself is false. The implication has not been proven false. The whole thing p→q is still true.
You’ll understand it better with an example from Rosen’s book :
A useful way to understand the truth value of a conditional statement is to think of an obligation or a contract. For example, the pledge many politicians make when running for office is
“If I am elected, then I will lower taxes.”
If the politician is elected, voters would expect this politician to lower taxes. Furthermore, if the politician is not elected, then voters will not have any expectation that this person will lower taxes, although the person may have sufficient influence to cause those in power to lower taxes. It is only when the politician is elected but does not lower taxes that voters can say that the politician has broken the campaign pledge. This last scenario corresponds to the case when p is true but q is false in p → q.
Similarly, consider a statement that a professor might make: “If you get 100% on the final, then you will get an A.”
If you manage to get a 100% on the final, then you would expect to receive an A. If you do not get 100% you may or may not receive an A depending on other factors. However, if you do get
100%, but the professor does not give you an A, you will feel cheated.